Quantum Backgammon

One of the ways to maintain interest in a competition that might otherwise be a somewhat monotonous series of individual games is to make some of those games more valuable than others. Backgammon does this in a really ingenious way.

The backgammon boom of the 1930s is often attributed to the invention of the doubling cube, a device that causes some games to be more valuable than others, but does so in a way that opens new avenues for skill.

Backgammon has always had a limited way of making some games worth more than others. Gammons (double games) are reasonably common. Backgammons (triple games) are quite rare, but not unknown. But the main way that backgammon raises the stakes on some games is with the use of the doubling cube.

The doubling cube is a separate die, often rather larger than the others, which is numbered with the powers of two: 2, 4, 8, 16, 32, and 64. It starts in the center of the board, showing 64 (which really means 1). When it’s in the center, before throwing the dice, either player may offer to turn the cube to 2. The other player may accept the cube, which means that the game will continue, but be worth twice as much as before – two dollars, or two points, or whatever. Or the player may resign and decline the cube, conceding one point (or dollar, or whatever). After that, the game may be redoubled to 4, re-redoubled to 8, and so forth. But subsequent doubles happen only at the instigation of the player who accepted the previous double.

The doubling cube improves the game in several ways. It ends a lot of relatively uninteresting games early. It raises the stakes on the interesting games that remain, and offers a player who’s behind in a match or a session played for money to catch up quickly. And it adds a deep layer of skill to the game. Understanding the proper use of the doubling cube is regarded by expert players as the most difficult aspect of the game.

The effect of the doubling cube on fairness (C) is not entirely clear. In one respect, it makes the game much less fair because it can magnify the element of chance, sometimes causing a single throw of the dice to determine an entire match. But, because it requires so much skill to use well, it might also tend to reward superior play.

A recent match I played illustrates the dramatic effect the cube can exert on a match. We were playing a 25-point match, which is as long a match as is usually played – most backgammon scoreboards only go up to 25 points. Such a match generally takes between four and five hours.

After a bit more than two hours, and a long series of games played for one or two points, the score stood at 14 to 13 in my favor. In the next game, I accepted a 2-cube, and then redoubled to 4 when the game turned decidedly to my advantage. I was a heavy favorite to win the 4 points, extending my lead to 18-13, which would have made me a distinct favorite to reach 25. But a single horrible throw of the dice (4-4) caused my position to collapse. and a series of poor throws late in the game cause me not only to lose, but to be backgammoned. That’s 12 points for my opponent, just enough for him to win.

Is this a fair result? Certainly, in the fairness (A) aspect. It’s hard to argue, however, that it was fair with respect to fairness (C). Whether or not the outcome should be considered fair, the drama made the match much more interesting to play, and for our one spectator to watch. If such things could not happen, it’s unlikely that the two of us would have been playing such a match at all.